A Helson matrix with explicit eigenvalue asymptotics

A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries { a ( j k ) } for j , k ≥ 1 . Here the ( j , k ) 'th term depends on the product jk. We study a self-adjoint Helson matrix for a particular sequence a ( j ) = ( j log ⁡ j ( log ⁡ log ⁡ j ) α ) ) − 1 , j ≥ 3 , where α > 0 , and prove that it is compact and that its eigenvalues obey the asymptotics λ n ∼ ϰ ( α ) / n α as n → ∞ , with an explicit constant ϰ ( α ) . We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.